Problem: Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 183 \times 193$ is divided by $5$.
Explanation: First, we use the property $a \equiv b \pmod{m}$ implies $ac \equiv bc \pmod{m}$.

Since all numbers with units digit $3$ have a remainder of $3$ when divided by $5$ and there are $20$ numbers,  $$3 \times 13 \times 23 \times 33 \times \ldots \times 183 \times 193 \equiv 3^{20} \pmod{5}.$$Next, we also use the property $a \equiv b \pmod{m}$ implies $a^c \equiv b^c \pmod{m}$.

Since $3^4 \equiv 81 \equiv 1 \pmod5$, and $3^{20} = (3^4)^5$, then $3^{20} \equiv 1^5 \equiv \boxed{1} \pmod{5}$.